BE (2nd Semester)
Examination April/May, 2009
1. (a) State the De-Moiver’s theorem.
(b) Find all the roots of the equation
(i) cos z=2
(ii) tanh z=2
(c) Separate sin-1 (cosᶿ + I sinᶿ) into real and imaginary parts, where ө is a positive acute angle.
(d) Sum the series:
n sin a +n (n+1) sin 2an (n+1) (n+2) sin 3a+………….∞
2. (a) Explain briefly the method of variation of parameters.
(D2+2) y=x2e3X+eX cos2x.
(c) Solve the equation:
(d) Solve the simultaneous equation:
dx +2y=et and dy– 2x=e-t
3. (a) Write the relation between Beta and Gamma function.
(b) Evaluate the integral by changing the order of integration.
ʃ ʃ y2 dxdy
0 √ax √y4– a2x2
ʃ ʃ e—(x2+y2) dxdy
by changing iro polar co-ordinates.
Hence show that:
ʃ e—x2 dx= √∏
(d) Find by double integration, the area lying between the parabola
Y=4x-x2 and the line y = x.
4. (a) State the Green’s theorem in the plane.
(b) Prove that:
A.V (B.V 1)=3(A.R)(B.R.) _ A.B
r r5 r3
Where A and B is constant vectors.
R=xI + yJ + K and r = √x2 + y2 +z2
(c) A vector field is given by:
F=(x2 – y2 + x)I – (2xy +y)J
Show that the Field is irrotational and find it’s scalar potential. Hence evaluate the line integral
from (1,2) to (2,1).
(d) Verify stokes theorem for the vector field
F=(2x – y) I – yz2J – y2zK on the upper half surface of x2 + y1 + z2= 1 bounded by its projection on the x y palne.
5. (a) Write the relation between roots and coefficient of the equation.
a0xn + a1xn-1 + a2xn-2 +……….+ an-1x + an = 0
(b) If aβϒ be the roots of x3 + px + q = 0
(i) a3 + β3 +ϒ3 = 5 aβϒ ∑ aβ
(ii) 3∑a2∑a5= 5∑ a∑ a4
(c) Show that the equation x4 – 10x3 + 23x2 – 6x ─15= 0 can be transformed into reciprocal equation by diminishing the roots by 2. Hence solve the equation.
(d) Solve the cardon’s method, the equation:
27x3 + 54x2 + 198x – 73= 0